A router by definition routes - it moves packets between networks.
A switch is what you use for a network, it passes packets between devices on the local network - the LAN. Switches often have the ability to support VLANs. Ports are allocated to a VLAN and cannot communicate with ports allocated to different VLANs, which effectively creates multiple virtual LANs (vlans).
A "layer 3" switch, is a switch that can route between VLANs. So it is both a switch, and a router. The routing aspect of a layer 3 switch is usually limited, and not intended to be used as a WAN router for example. Its role is usually to route packets between the VLANs it has configured.
Some routers, such as domestic routers, come with switchports. So this is effectively a router with a switch.
These last type of routers can definitely support a LAN. However, it only has a few switchports, so couldn't support a large LAN.
The article is really referring to the absolutes of the term router, which is blurred in reality to target different needs.